A202868 Array: row n shows the coefficients of the characteristic polynomial of the n-th principal submatrix of the symmetric matrix A115216; by antidiagonals.

1, -1, 1, -6, 1, 1, -11, 27, -1, 1, -16, 78, -112, 1, 1, -21, 154, -458, 453, -1, 1, -26, 255, -1164, 2431, -1818, 1, 1, -31, 381, -2355, 7635, -12141, 7279, -1, 1, -36, 532, -4156, 18390, -45660, 58260, -29124, 1, 1, -41, 708, -6692, 37646, -128190

Offset

1,4

Comments

Let p(n)=p(n,x) be the characteristic polynomial of the n-th principal submatrix. The zeros of p(n) are positive, and they interlace the zeros of p(n+1).

Links

Table of n, a(n) for n=1..50.

S.-G. Hwang, Cauchy's interlace theorem for eigenvalues of Hermitian matrices, American Mathematical Monthly 111 (2004) 157-159.

A. Mercer and P. Mercer, Cauchy's interlace theorem and lower bounds for the spectral radius, International Journal of Mathematics and Mathematical Sciences 23, no. 8 (2000) 563-566.

Example

The 1st principal submatrix (ps) of A115216 is {{1}} (using Mathematica matrix notation), with p(1)=1-x and zero-set {1}.
...
The 2nd ps is {{1,2},{2,5}}, with p(2)=1-6x+x^2 and zero-set {0.171..., 5.828...}.
...
The 3rd ps is {{1,2,4},{2,5,10},{4,10,21}}, with p(3)=1-30x+57x^2-x^3 and zero-set {0.136..., 0.276..., 2.587...}.
...
Top of the array:
1...-1
1...-6....1
1...-11...27...-1
1...-16...78...-112...1

Mathematica

f[k_] := 2^(k - 1);
U[n_] := NestList[Most[Prepend[#, 0]] &, #, Length[#] - 1] &[Table[f[k], {k, 1, n}]];
L[n_] := Transpose[U[n]];
F[n_] := CharacteristicPolynomial[L[n].U[n], x];
c[n_] := CoefficientList[F[n], x]
TableForm[Flatten[Table[F[n], {n, 1, 10}]]]
Table[c[n], {n, 1, 12}]
Flatten[%]  (* A202868 sequence *)
TableForm[Table[c[n], {n, 1, 10}]]  (* A202868 array *)
Table[(F[k] /. x -> -1), {k, 1, 30}]   (* A154626 *)
Table[(F[k] /. x -> 1), {k, 1, 30}]    (* A058922 *)

Crossrefs

Cf. A115216, A202605.

Sequence in context: A081579 A295707 A146772 * A202877 A174124 A174345

Adjacent sequences: A202865 A202866 A202867 * A202869 A202870 A202871

Keyword

tabl,sign

Author

Clark Kimberling, Dec 26 2011

Status

approved